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A270749 (r,1)-greedy sequence, where r(k) = k/log(k+1).

%I A270749 #13 Mar 06 2025 08:17:59
%S A270749 2,7,117,28231,934841727,1391154929853413822,
%T A270749 3358221400639080017571595039208647108,
%U A270749 84149630763494298099512446622134485046922136023978562834130778814722933257
%N A270749 (r,1)-greedy sequence, where r(k) = k/log(k+1).
%C A270749 Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers.  Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x.  The sequence (a(n)) is the (r,x)-greedy sequence.  We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x.  See A270744 for a guide to related sequences.
%F A270749 a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).
%F A270749 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%e A270749 a(1) = ceiling(r(1)) = ceiling(1/log(2)) = ceiling(1.442...) = 2;
%e A270749 a(2) = ceiling(r(2)/(1 - r(1)/2)) = 7;
%e A270749 a(3) = ceiling(r(3)/(1 - r(1)/2 - r(2)/7)) = 117.
%e A270749 The first 3 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.721..., 0.981..., 0.99991...
%t A270749 $MaxExtraPrecision = Infinity; z = 16;
%t A270749 r[k_] := N[k/Log[k + 1], 1000]; f[x_, 0] = x;
%t A270749 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270749 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270749 x = 1; Table[n[x, k], {k, 1, z}]
%t A270749 N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]
%Y A270749 Cf. A001620, A270744.
%K A270749 nonn,easy
%O A270749 1,1
%A A270749 _Clark Kimberling_, Apr 09 2016